Or...The best thing about this is watching the students' reactions

I don't remember where I found this video...I'm sure someone posted it on #Mtbos at some point. If you don't get anything out of this blog post other than the pure enjoyment of watching this video...I'll feel like I've done my part.

I asked twitter if anyone had ever created a combining like terms lesson using this video and Annie replied with this blog post.

The Lesson

I wanted to try to elaborate on her ideas a bit and this is what I came up with.

I showed the video. This was the best part. The kids were so frustrated by the end! I did stop the video right at the part where the food gets brought out to save a few seconds.

I asked the kids what they noticed/wondered. Here were some responses:

I noticed that guy's going to get really fat.

I noticed that he kept ordering the same things over and over again, like a small coke and a small coke.

I noticed they kept saying the same order over and over again instead of saying it shorter.

I wanted to bang my head against the wall the 4th time they said the order.

I noticed there were different ways to make $11 at the end.

If I were that lady I'd tell the guy to go somewhere else.

They were talking really fast!

After asking the question presented, I passed out one copy of the original order per group in a plastic sleeve. Students spent about 5 minutes making the order more efficient.

I found it interesting that some groups decided to add all the cheeseburgers together and all the hamburgers together. They said they'd just tell the guy they don't do special orders. :)

I thought it would be fun to make the connection to how In-n-Out reads back their orders. If you've never been, they combine all like items together: "1 hamburgers, 2 cheeseburger, 3 fries and 3 drinks."

This is kind of a West Coast thing...so if you don't have In-n-Out, I'm not sure this would be relevant to your students.

After a quick discussion on the relevance to combining like terms, we took a few notes. and did some practice.

My Thoughts

This lesson was lots of fun. It really hooked the kids and it seems like they made the connection to combining like terms.

They wasted a bit of time just reading the script right after I passed it out. This was just supposed to be a review lesson for my 8th graders so I didn't want to spend too much time on it which is why I only gave them 5 minutes to rewrite the script. Most groups did not completely finish, but they got the idea. If this were the first experience students had with combining like terms I would absolutely give them enough time to rewrite the script.

Even though they "hated" the video, every class asked me if I would post it on Google Classroom.

If I do combining like terms review again next year, I will most likely use the video again.

Or The last Math teacher to ride this train...

I've always been an advocate for hands on learning and students making observations to create their own rules, but for some reason I've always forcefully led students to one way of thinking about finding area of parallelograms, triangles and trapezoids. I'll say it was more of a "hands on lab" than a "discovery lesson."

I would give students some parallelograms, triangles and trapezoids and tell them what to do:

Parallelogram: Cut off this piece and make it into a rectangle! Whoa!Triangle: Make a copy, flip it over and match the sides to make a parallelogram! Whoa x 2!Trapezoid: Make a copy, flip it over and match the sides to make a parallelogram! Whoa x 3!

I'll admit that I still find these relationships between the shapes pretty cool and it's how I think about area to this day. However, I made it a goal to make discovery learning a bit more discovery this year. I wasn't going to lead the kids...I was going to let them have at it and see where it went. Let's just say it took me to places I've never been before.

Jump to the trapezoid section to get to the really interesting part!

Parallelograms

The parallelogram explorations went pretty according to plan. I started by giving students a set of parallelograms printed on grid paper and asked them to find the area. There were 3 distinct methods.1. Count squares and estimate partial squares. (Not very accurate or easy without grids.)2. Put rectangles around the triangular parts of the parallelogram and find the areas separately. (Not going to be useful when students are given the measurements.)3. Cut off the triangular piece on one side and fit it in the empty space of the parallelogram to make a rectangle. (Yay! Someone got it!)

I put a blank parallelogram on the board and asked students what measurements they would like in order to find the area. I gave them all they asked for. Then, I gave them a parallelogram with my choices of measurements and asked them to find the area.

Students realized the third method was the best (most were already using that method once we reached this point.)

Triangles

Triangles went a little different than expected, but ultimately we came up with the same results. I followed the same structure as parallelograms. Here were their methods:1. Count the squares and estimate partial squares (only 1 or 2 students per class stuck to this method.)2. Cut the triangle in half and fit the piece in the missing spot (only worked for isosceles triangles.)3. Split the triangle into 2 right triangles and find the areas of each separately. (This was the favorite method.)4. Double the triangle and make it into a parallelogram, make the parallelogram into a rectangle and then find half that area. (Most students did this with the right triangle, but only some with the other types.)

When we got to the part of the lesson where I gave the dimensions and asked students to find the area, many were stumped. They REALLY wanted to know the 2 different lengths of the bases of the right triangles. Luckily, one student from each class determined that it doesn't matter. If you find the areas of the 2 right triangles by taking half of one rectangle and half of the other rectangle that's the same as taking half of the whole rectangle to find the area of the whole triangle. (Yay!!)

Ergo, base times height divided by 2. Got the same equation with a different than my usual method.

Trapezoids

This part right here is the entire reason for me taking the time to write this blog post. I was absolutely floored with what came out of this activity.

When I gave the trapezoids on grid paper all I heard was "this is so easy!" "I love trapezoids!" And I kept thinking...ha...just wait! Every student solved them the same way by splitting it into 3 parts: Two triangles and a rectangle in the middle.

I wasn't surprised this was the go-to method, but I knew they'd struggle when I gave them the dimensions of a trapezoid without squares...because I was NOT going to give them the larger base split into parts. I was not prepared with what they came up with.

Looking at this now, it's incredibly obvious, but I had never thought of finding area of trapezoids in this way. It even results in a different formula:Short base * height + (long base - short base) * height / 2

I know this formula doesn't match the formula they will be given on any assessment. I don't know that any of the students will even remember the formula they created. I don't know how many students will be able to reproduce what we did a year from now. But I know I let my kids roll with something and they came up with something wonderful.

Every year I have students that struggle with finding the measures of missing angles. Simple problems like this, they have no trouble with:

It's when the line and angle system gets more complex that students begin to get frustrated:

They struggle with finding those complementary, supplementary and vertical angles. I've always told students to turn the paper and cover up extra information, but many cannot ignore the extra to focus on what's important. So, I made a few things to help.

Now students can ignore the extra and focus on what's important in solving for the missing angles.

About a year ago, Saturday Morning Breakfast Cereal published the following comic:

Being familiar with the rice on a chessboard problem, I found it rather hilarious...so of course I wanted to share that hilarity with my students. I showed them just the first 4 squares and asked them why the boy would choose to take the pennies. The student responses were anything from "he's an idiot" or "he's not very good at math" to "he knows his grandpa won't really pay for his college, so he'd rather get some money than nothing."

That's when I introduced the rice problem and explained that they boy had probably heard of this problem as well. So, we started to figure out just how much he would get if the pennies doubled on each square. The students, armed with pencil, paper and calculators quickly set to work. They probably would have finished the problem (they were pretty into it) but this was just supposed to be a warm up, so I broke out a spreadsheet to quickly calculate.​

Students were naturally curious about what the E+ meant which led to a quick discussion on scientific notation. Even students who didn't understand realized that the total amount of the pennies on the board would be way more than the cost of a college education!

This year when it came time to actually teach scientific notation to my Advanced 7/8 class, I remembered this moment and purposely chose this activity to intro the topic. I was a little more methodical to my approach, but it was pretty similar and worked out incredibly well:

Again, the students were shocked at how quickly the pennies increased. Since it was my advanced class, we discussed what the graph of the relationship would look like and the resulting exponential equation.

The students were using calculators to explore the problem and some got to the scientific notation form quicker than others. This was the point that I decided to use the spreadsheet for all to see.

When the last panel is finally revealed, students inevitably want to know where in the world the $1.27 came from...I let them figure it out.

I wanted to share my struggles with my Expressions and Equations unit. I'm finding I have so many good ways to model solving equations, but I haven't yet found that one perfect method. My guess is there isn't one...instead, students can be introduced to a variety of models that can be woven together into a coherent tapestry of algebraic understanding. Problem is....I'm not a great sewer.

Here is a run down of my experiences (the pros and cons) of the different strategies I've used and my experiences.

Algebra Tiles: Algebra tiles incorporated with the "balance scale" method has always been my go-to for solving equations. It's tried and true and I'm comfortable with it. Before introducing equations, algebra tiles are great for modeling simplifying expressions. Students can visualize that 2(x+4) is actually 2 groups of x+4 which is 2 x's and 8. It starts getting a little iffy when tiles are used for problems such as -2(2x-4) and (x+5)-(-2x-3), but for the most part students are successful. For me, algebra tiles start to fall apart when introducing fractions and decimals into equations.

Pros

Great for transitioning into a more algebraic approach. Students see the method of what you do to both sides to isolate the variable. (Adding 2 red tiles to zero out 2 yellow tiles is the same as adding negative 2.)

Great for visualizing positive and negatives values in equations.

Great for seeing different paths to solve. For example, in the problem 3(x+4)=12, you can either combine like terms to get 3x +12 in which the first step is to distribute or you can divide the algebra tiles into groups of 3 and notice that x+4 is the same as 4 in which the first step is to divide both sides of the equation by 3.

Cons

Not overly useful for equations with decimals and fractions, although it is possible with a dry erase marker and as long as the fraction partitions aren't too large.

Students have a hard time modeling subtraction of expressions and often model addition instead.

Here's an example of a couple students trying to grapple through using algebra tiles to solve a problem with fractions.

Working Backwards StrategyFor the first time last year I used the "working backwards" strategy to solve equations. This stemmed from the amazing MARS task Building and Solving Linear Equations. This task tells the story of x and in doing so, students create some pretty complex equations. It then shows how we can walk the story backwards to determine the value of x. The students were extremely successful with solving equations this way and even my students that typically struggle were solving complex equations with ease. I loved it! However, I realized this strategy falls apart when you have variables on both sides of the equation or 2 variables on the same side. Because of this major limitation, I chose not to use this method this year.

Pros

Solidifies order of operations in equations. Students really see what comes first, second, etc...

Allowed for students (even my lowest) to be successful in solving pretty complex equations with 1 variable.

Cons

This strategy is limited to equations with 1 variable. I don't see a way for it to work with a variable on both sides or even more than one variable on the same side.

Double Number Line (Clothesline)
I'll admit, I'm a little obsessed with number lines this year. It hasn't been without it's bumps and I've learned a lot about how to implement them next year, but I've used number lines with just about everything: Operations with rational numbers, proportions, and then I tried double number lines to solve equations. It didn't go great, but I think that was due to my implementation rather than the method itself.

This year, I used Algebra tiles to start talking about expressions and equations. When it was time to incorporate fractions and decimals, I decided to try the double clothes line I first saw from Andrew Stadel, Jon Orr and Matt Vaudrey. The problem was that the students didn't have enough time to get comfortable with expressions on the clothes line before I started throwing in negative variables and non-integer numbers. In an hour long class period, they got 3 problems completed. It was actually a great problem solving opportunity, but didn't lend itself well to the actual method for solving. The students quickly went back to the standard method of isolating the variable in which they felt more comfortable.

Pros

Can be used for pretty much any type of equation.

Helps solidify number sense while solving equations. Students are actually thinking about the numbers and their relationship to the other numbers while placing them on the clothes line and not just going through a routine.

Gets the kids up and is kinda fun!

Cons

Much more time consuming than other methods, in my experience. This could change for classes that spent more time solving equations on a double number line before incorporating fractions. However, just the process filling out the number tents is time consuming.

Students didn't naturally gravitate to wanting to solve the problems this way. When given a choice, they chose algebra tiles. Again, this could be an implementation issue.

I have learned so much on a personal level about solving equations since joining Mtbos. It's been fun for me to play around with different methods and see how they connect. I'm not sure, however, this has been beneficial to my students. I definitely need to rethink things for next year. Is it best to stick to one method? Show all of the methods and let students choose? How much time do I spend on the concrete methods before moving into more abstract problems in which the models start to fall apart? I have vague answers to each of those questions, but I wonder what others do that is successful. Feel free to comment or hit me up on twitter if you have the answers. :)

Anyone who currently teaches Middle School students most likely knows about the bottle flipping phenomenon that took the country by storm.

There's been quite a bit of press about how it is being banned in schools and how teachers find it annoying. Do I find it annoying? Yes. Do I think it should be banned at school? No. It's a fun activity students enjoy. Why squelch it?

On the flip side, some teachers automatically looked for a way to incorporate bottle flipping into their classrooms. Dan Meyer wrote about his attempt at this here. I left my thoughts in his comments about how I struggled with the idea of teachers trying to take everything that is fun for students and incorporating it into school. I wasn't going to touch bottle flipping....I was going to leave it fun for the kids.

Well, after following all of the twitter conversations and Jon Orr's blog post (Flippity Flip, Bottle Flip!) I decided to give it a shot....and what better day to flip bottles than the day after a Monday Halloween?

I started by showing the following video. (Note: There is an f-bomb that needs to be muted at about 2:08)

I asked what math they saw in the video. Students replied with the following:

How much water needs to be in the bottle to flip it?

Does the size of the bottle matter?

How high are the flips?

How many times does the bottle flip around based on the height of the flip?

What is the probability of the last flip in the video actually happening?

How many flips can you land per minute? (This one was from my 6th grade class. I asked them what rates they could find from the video.)

How Much Water is Best?

We compiled all data from my 7th grade classes into one graph. Here is the overlay of the individual scores.

The last slide of the Desmos activity sums the data and plots it....Where's the 30% dot???

A couple thoughts on the activity:

One student suggested we use straws to drink out the water...that keeps the water bottle upright to more easily view the water level.

I went back and forth on whether to measure height of the water level or volume of water. Due to the curve of the top of the water bottle, the 10% increments of height doesn't equal the same increments in volume. I"m unsure which way is best, but since most kids aren't going to have measuring cups with them when they're flipping, I suppose the way we did it is fine.

There are definitely environmental implications to this activity. I was sure to have students share water bottles and we recycled them after. I think having a discussion with your students about decreasing their footprint is a good idea here.

Who's the Best Flipper?

We really wanted our 6th graders to be able to flip bottles as well, but we haven't discussed percentages with them. However, we're right in the middle of unit rates, so we decided to answer the question of "Who is the best flipper?" by finding every student's flips per minute rate.

What's Next?

Well, I don't think I destroyed anyone's love for bottle flipping by mathifying it. The kids had a blast...and they did some great math in the meantime.

A few questions came up that we might try to answer later in the year:

Is the best percent filled line the same for any sized water bottle?

Do different people's flip styles have any affect on the water level that works best?

Some students asked for a new bottle after the bottoms of theirs started wearing down. They said the bottle wouldn't land as well. However, I was observing that used water bottles seemed to land better than new ones. Could be something to test!

This past week my 7th grade class was starting on multiplying rational numbers….which of course means...it’s time to see what they remember from 6th grade. I went back and forth on how to present the review...do I just remind them of the algorithm? Do I do a quick “hey, remember this model” then jump to the algorithm? Do I have them reinvent the algorithm? Due to the fact that I knew they modeled multiplying fractions last year and that we’ve been having some hard core discussions about what our operations mean, I determined that I would just see what they could do.

I had students working in groups of 3-4 and in the middle of the table was a plethora of manipulatives...a sampling of pretty much everything I have in my cabinets. I gave each group a note card with a fraction multiplication problem and the following prompt: "Use the stuff at your table to model your multiplication problem as many different ways as you can."

I'll be honest, I had pretty high expectations of what I'd get. We've been doing What's Your Story every week since the start of school and we've had many conversations about what a fraction of a group could look like. Unfortunately, this is what most groups did:

I decided to take a step back and discuss as a class what 1/2 x 1/2 means...1/2 of a group of 1/2. I asked each group to show a way to model it. Things went a little better, but I'll be honest...after the first flop, I was feeling defeated...I just wasn't as excited as I was at the beginning of the lesson. I was having that initial internal struggle with myself:

Do I spend time solidifying what multiplying fractions means before adding in decimals and negatives?

Do I say "screw it" and just remind them, "Hey! Remember that you just multiply across? Yes, the easiest thing you ever have to do with fractions can be completely convoluted and confusing by using these models I'm trying to force on you in order to glean some understanding of what's happening to the numbers???"

How soon do I start throwing in mixed numbers?

It's almost November and I'm STILL on Unit 1! WTF am I doing???

Long story short, I decide to put the manipulatives away and try our hand at modeling on paper using grids. We did some problems together and with about 5 minutes left in class I gave them a ticket out the door. Results were pretty much what I expected. About 1/3 of them tried to model and failed, 1/3 were able to draw the model and 1/3 didn't even try the model and just multiplied across.

I'd like to say that after that first failure I was able to change things up for the other classes in a way that made things wonderful for them and me...unfortunately, even with the minor changes I made to the lesson (like starting with the 1/2 x 1/2 conversation) the results were pretty much the same. My lower students were getting confused with the model. The higher students were annoyed that I was even asking them to model. It was just, overall, not a great math day.

Ideally, this is the part of the blog where I'd talk about what I'd do differently...how I'd adjust my lesson for next year...but at this point my brain isn't ready to process that. Right now I'm trying to think about what went so wrong...what did I do differently from years past to make this lesson flop so hard. If I ever figure it out, I'll let you know.

Today we reviewed converting fractions into decimals using the following worksheet. (Thanks to Illustrative Mathematics for the last two questions.)

While walking around, I saw a pair of students who said 1/2 was equal to 1.2. This is a pretty common error in 7th grade. Here's a summary of the conversation we had (Eventually I'll remember to use my voice recorder on my phone for better accuracy.):

Me: Can you show me where 1/2 is on a number line?Student 1: It's half way between 0 and 1.Me: Where would 1.2 be on a number line?Student 2: It's after the 1. Between the 1 and the 2.Me: So, can 1.2 be equal to 1/2?

(At this point, there was a bit of discussion between the girls about whether or not they placed the numbers on the number line correctly. After they decided they were correctly placed, they determined that 1/2 could not equal 1.2.)

S2: Oh yeah, if it was 1.2, there would be a 1 in front of the fraction, so it has to be 0 point something.Me: So, what do you think 1/2 is as a decimal?S1: 0.2 Me: Why?S2: Because it has to be a 0 before the decimal and there's a 2.

All right...back to the number line...

Me: Can you show me what that would look like on the number line?S2: If I split this in to 4 equal pieces, 1/2 is at the second line, so that would be 0.2

Me: Ok (I point to the other marks.) Can you tell me what these would be, then?S2: (Points to the 1st mark) This would be 0.1. (Points to the 3rd mark) This would be 0.3. (Points to the 4th mark) This would be 0.4

​(At this point, I've basically sat back and let the girls go...hoping for a glimmer of light.)

S1: That's not right.​S2: Yeah, 0.4 is not 1.....Wait....we need to split this in to 10 pieces.

(Oh, thank goodness...)

Me: Can you explain why?S2: Because that's what the decimal place means. That's the tenths place.​Me: Ok, so try that.

S1: So, it's 0.5?Me: Does that make sense?S2: Yeah, because it's like 50 cents is half of a dollar.​S1: Oh yeah.Me: Awesome.

We had a few more struggles along the way, but after that first interaction, their desire to just replace the fraction line with the decimal stopped. They actually starting thinking about what the denominator meant in relation to the decimal equivalence.

Making the Jigsaw

Tarsia is a free software from Hermitech Labs that creates customized mathematical (or otherwise) jigsaw puzzles. I made one for students to review adding and subtracting rational numbers.

The user interface is super easy. Just type in a problem and it's answer for each card. It has an equation editor for mathematical expressions. It mixes up the jigsaw pieces for you and even allows you to print off a table of answers or a pdf of what the finished puzzle looks like.

Here's a page of the puzzle pieces (it printed out on 2 pages) and the answer key. I didn't save the file before my computer decided it needed to install updates over lunch...so...this is the only copy I have right now. :)

Digital Number Line

We've been working with adding and subtracting rational numbers on an interactive number line using a Google App from The Math Learning Center.

It's great because students can see the jumps and it really solidifies that transfer of what they learned about integers on the number line to what they already know about fractions and decimals.

This concept is NOT easy, but I have seen more successes using this number line than I have with memorizing integer rules in the past.

Using the number line app to solve a problem.

Thoughts About The Day

I have to say, I'm impressed with how long the students stuck with this. The puzzle I gave them had 16 matches...so they had to answer 16 problems. When I was making it, an hour seemed like plenty of time, but in the moment, it was apparent how much time and thought they were taking for each problem. Only a handful of pairs got finished by the end of class and we'll finish it up tomorrow. If I do this puzzle again (with the same exposure to solving the problems they've had) I'll do a smaller one for sure!

​Overall, they seemed to love the puzzle. They were actively engaged the whole time, were getting incredibly excited when they found matches and even asked if they could come in for study hall to keep working on it. One girl that finished right before class ended said, "We're done! That was fun...but man, my brain hurts." I guess I can't ask for a better reaction than that.

Double Number Line

In my 7/8 Math class we're expected to cover 7th and 8th grade standards in 1 year. Last year I flew quickly through the 7th grade ratio and proportion strategies in order to allow more time for the 8th grade Linear Expressions unit. Unfortunately, this kind of back fired because the experiences they needed in understanding how ratios work would have made the 8th grade standards come much quicker.

This year, I'm trying to remedy that. We still need to move quickly, but I'm trying to incorporate lots of strategies at one time and how they relate to each other.

Today, we started with a ratio problem using a double number line. I wrote up my plan to implement it here.

I used dry erase tents that I created by putting strips of packing tape on construction paper. They worked pretty well!

I used single strips of tape and trimmed off the edges. To the right are finished tents. The final picture is 2 pieces of tape together to make a larger tent.

I had students creating the number lines on whiteboards at their desks. I feel like I spent a little too much time on the activity and the students were getting a bit antsy. I think next time I wouldn't worry about getting all of the groups a chance at the number line or I'll switch to a different problem. It could also be that the problems wasn't challenging enough for these kids, but I'm still glad I did the activity because it's a good visual and they have a reference for how to solve ratio problems if they ever get stuck.

When I pointed to the ratio 35:1, I asked the question, "Does anyone know what this is?" The student I called on said, "The lowest it can go." I was SO EXCITED for his mistake! It allowed us to have a conversation about whether or not that was the lowest the ratio could go. Also, would it make sense to go in to the negatives? They had some fun trying to come up with situations that would result in the ratio being negative. (No...driving backwards is still positive miles! I need to show them the Ferris Beuller clip!)

Round the World

This is an activity I haven't done in a long time. I have 5 groups in the class so I put up 5 posters with problems. Each group had 1 color of marker and 3.5 minutes to start solving the problem on the poster. After the time is up, they switch posters. At the new poster they have 2 choices. They can either continue solving the problem using the method the other group used or they can start solving it their own method. We continue going around until every group has hit every poster.Here are my thoughts...

I honestly don't know if this is a good or bad strategy in math. I love the idea of students looking at other people's work, trying to decipher it and then continuing it...but I don't know if the time limit in this case is harmful. I don't feel like it is because they get to come back to the poster later and see what the results were.

For this class, I think 3.5 minutes was too much time (at least for the problems that were presented...most were DOK 1 & 2.) . Almost all groups were able to solve the problems in the time allowed with time to spare.

At the start of the activity, some groups spent very little time looking at the other strategies and just wanted to solve the problem their way...even if it meant duplicating a method that was already on the paper. Hopefully if we do this again, they'll have a better idea of what is expected.

I think it's just because this is an "advanced level" class, but the posters were all very arithmetic heavy...it was the last session and a group was trying to come up with a new strategy and I heard someone say, "No one did any pictures." I was like, "No! They didn't! You should do that!" Unfortunately they didn't have time to finish, but at least we got to see that that method COULD solve that problem!

I gave them time at the end to walk around to all the posters and see the methods others used to solve the problems. This allowed me to get an idea of what they know how to do as well as give the students some new ideas for how to solve these types of problems. Overall, I fell like it was a success.